How do I turn my PNG Image into a mesh object?
If you want in to be well done import the image as a viewport background image and manually trace over it using a bezier curve object.

If you are more comfortable working with applications using an illustration workflow like Inkscape, you can manually trace it using a pen tool and then export/import as an SVG.

Alternatively, you you want a faster method Inkscape also has a builtin image tracer that can trace a raster file and create the corresponding vector automatically but the quality some times leaves a lot to be desired and your mileage may vary.